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Homework Help: Number of real variables required to construct the most general 2by2 unitary matrix

  1. Aug 28, 2008 #1
    1. The problem statement, all variables and given/known data

    how many real variables would be required to construct a most general 2by2 unitary matrix?

    2. Relevant equations

    a unitary matrix U is one for which the U(U hermitian) = identity matrix or (U hermitian)U = identity matrix

    3. The attempt at a solution

    first i wrote the unitary matrix as {{a1+ib1,a2+ib2},{a3+ib3,a4+ib4}}, where 'i' is square root of -1. using the condition U(U hermitian) = identity matrix, i get four independent equations. thus, i should expect the number of real variables required as 8 - 4 (number of constraints) = 4. but if i consider the definition (U hermitian)U = identity matrix, i get another set of four equations. Does that mean that the number of real variables required is 8-8 =0?

    there is another way to attack the problem. a hermitian matrix must have real diagonal elements. the (1,2) element must be the complex conjugate of the (2,1) element and hence, i need 4 real variables to construct the most general 2by2 hermitian matrix. since a unitary matrix can be written as exp(iH) where H is a hermitian matrix, does this also indicate that i would need 4 variables for a unitary matrix as well?
     
    Last edited: Aug 28, 2008
  2. jcsd
  3. Aug 28, 2008 #2

    Dick

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    Re: number of real variables required to construct the most general 2by2 unitary matr

    You are counting correctly, except (U*)U=I and U(U*)=I are not independent sets of equations. If one holds the other automatically holds. There are 4 real parameters describing a 2x2 unitary matrix.
     
  4. Sep 2, 2008 #3
    Re: number of real variables required to construct the most general 2by2 unitary matr

    thank you for your assistance... i later on figured out that it must be true since when i impose the condition that the determinant of the matrix is 1, then i end up getting SU2 group, to specify which i need three parameters... so, without the constraint of the determinant being 1, i must need 4 parameters
     
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